Signature matrix algebras and bipartite graphs

A special class of matrix algebras, the rc-signature algebras, naturally emerged as a result of the study of a Multiplicative Decomposition Property of matrices (a multiplicative analogue of the Riesz Decomposition Property in ordered vector spaces). This note is devoted to the study of a tractable subclass of these algebras. It is proven that a necessary and sufficient condition for two such algebras to be isomorphic is the simultaneous permutation-similarity between the members of the algebras. There is a one-to-one correspondence between the signature algebras and the bipartite graphs that respects the isomorphism between the algebras and the strict isomorphism between the bipartite graphs.